40 research outputs found
Combinatorics and topology of toric arrangements defined by root systems
Given the toric (or toral) arrangement defined by a root system , we
describe the poset of its layers (connected components of intersections) and we
count its elements. Indeed we show how to reduce to zero-dimensional layers,
and in this case we provide an explicit formula involving the maximal
subdiagrams of the affine Dynkin diagram of . Then we compute the Euler
characteristic and the Poincare' polynomial of the complement of the
arrangement, which is the set of regular points of the torus.Comment: 20 pages. Updated version of a paper published in December 200
Geometric realizations and duality for Dahmen-Micchelli modules and De Concini-Procesi-Vergne modules
We give an algebraic description of several modules and algebras related to
the vector partition function, and we prove that they can be realized as the
equivariant K-theory of some manifolds that have a nice combinatorial
description. We also propose a more natural and general notion of duality
between these modules, which corresponds to a Poincar\'e duality-type
correspondence for equivariant K-theory.Comment: Final version, to appear on Discrete and Computational Geometr
A Tutte polynomial for toric arrangements
We introduce a multiplicity Tutte polynomial M(x,y), with applications to
zonotopes and toric arrangements. We prove that M(x,y) satisfies a
deletion-restriction recurrence and has positive coefficients. The
characteristic polynomial and the Poincare' polynomial of a toric arrangement
are shown to be specializations of the associated polynomial M(x,y), likewise
the corresponding polynomials for a hyperplane arrangement are specializations
of the ordinary Tutte polynomial. Furthermore, M(1,y) is the Hilbert series of
the related discrete Dahmen-Micchelli space, while M(x,1) computes the volume
and the number of integral points of the associated zonotope.Comment: Final version, to appear on Transactions AMS. 28 pages, 4 picture
The multivariate arithmetic Tutte polynomial
We introduce an arithmetic version of the multivariate Tutte polynomial, and
(for representable arithmetic matroids) a quasi-polynomial that interpolates
between the two. A generalized Fortuin-Kasteleyn representation with
applications to arithmetic colorings and flows is obtained. We give a new and
more general proof of the positivity of the coefficients of the arithmetic
Tutte polynomial, and (in the representable case) a geometrical interpretation
of them.Comment: 21 page
The homotopy type of toric arrangements
A toric arrangement is a finite set of hypersurfaces in a complex torus, every hypersurface being the kernel of a character. In the present paper we build a CW-complex S homotopy equivalent to the arrangement complement ℜ x, with a combinatorial description similar to that of the well-known Salvetti complex. If the toric arrangement is defined by a Weyl group, we also provide an algebraic description, very handy for cohomology computations. In the last part we give a description in terms of tableaux for a toric arrangement of type à n appearing in robotics.Arrangement of hyperplanes, toric arrangements, CW complexes, Salvetti complex, Weyl groups, integer cohomology, Young Tableaux
The homotopy type of toric arrangements
A toric arrangement is a finite set of hypersurfaces in a complex torus,
every hypersurface being the kernel of a character. In the present paper we
build a CW-complex homotopy equivalent to the arrangement complement, with a
combinatorial description similar to that of the well-known Salvetti complex.
If the toric arrangement is defined by a Weyl group we also provide an
algebraic description, very handy for cohomology computations. In the last part
we give a description in terms of tableaux for a toric arrangement appearing in
robotics.Comment: To appear on J. of Pure and Appl. Algebra. 16 pages, 3 picture
Wonderful models for toric arrangements
We build a wonderful model for toric arrangements. We develop the "toric
analog" of the combinatorics of nested sets, which allows to define a family of
smooth open sets covering the model. In this way we prove that the model is
smooth, and we give a precise geometric and combinatorial description of the
normal crossing divisor.Comment: Final version, to appear on IMRN. 23 pages, 1 pictur